@ger21 described exactly how I would carry out the project with one exception, and that is the choice of router bit. I would also heed the advice of @gdonham1 and use a high feed rate in order to avoid melting. My choice of router bit is based on the width of the already existing grooves of which @Martyc wants to round the edges. A point cutting roundover bit would not apply a full 90° roundover in a single pass while an ovolo, a.k.a plunge roundover with a 1/4 inch small diameter would.Use a point cutting roundover bit, with a template and guide bushing.
Melting should not be an issue at all when cutting roundovers.